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A planning task is defined as a four tuple $(I, A, S, G)$. 
$I \in S$ is an initial state.  $A \in S \times S$ is the set of actions. 
$S$ is the set of states (also called state space) and $G \subseteq S$
is the set of goal states. To solve this planning task is to find a sequences of actions
$a_{i} \in A$ that can lead us from the initial state $I$ to a goal state $g \in G$. 

In this paper, we define a best-first search procedure 

A search procedure is that retrieves states from an open-list, 
expand it, check if it a goal and push it back to closed list. 
}

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To analysis the plateau. We introduce the following definitions. 
 For a planning problem and any state 
$s$ in the search space $S$ if it is not possible 
to satisfy the goal condition starting from $s$. 

For any state that is not a dead-end, The level of $s$ is defined as
the heuristic function value of $s$. We conveniently use $\infty$
to denote the heuristic values of those dead-end states. 
}